M.L.
When you're an adult and new in town, it's all right to talk to strangers. You have the common sense to know how to handle it. Anyone who makes you feel uncomfortable you can distance yourself from.
Everybody is timid sometimes, for various reasons. Don't let your background keep you from knowing some people and letting them know you.
When you go to the moms' group, wear a smile. (And clothing, of course.) Smile at *everyone* and say hello. A smile is actually a present, and most people don't get enough of that kind of present. Expect to discover some people who won't give you a glance or a kind word; it happens everywhere, and has nothing to do with you! If there's a greeter at the door, introduce yourself. Tell her you're new, and she might introduce you to others. That's what she's there for.
Church is usually a great place to make friends, because there are so many opportunities. Feel free to introduce yourself and let the person know you're new. At a small church, many people might realize right away that you're new! At a large church, folks may be thinking, "Is that a new lady or someone whose name I should already know?" and might be hesitant to say anything, so you can make them feel at ease by giving them the information they need.
Do the same with your neighbors. They will be glad to know who you are! They may be a little timid, too, but they're wondering what the new folks are like. You can ask your neighbors about good babysitters, too. Is there something you like to do for fun - photography, sewing, hiking? You could ask your neighbors if there are any organizations around devoted to your interest.
I would tend to be a bit more reserved at a park or other such public place, but I would smile - especially at the mothers with children.
Ask people about their all-time favorite subject, themselves! If you're at a church, ask also about the church, the service, the activities, the history of it; with the moms' group, ask about good shops, doctors and dentists, places in the area your children would like, things a family can do together - that sort of thing. Let them do the talking. They might, in turn, ask you about yourself; if they don't, don't worry about it - they're just not interested right now. That takes nothing away from you.
I have read that, by and large, strangers usually feel more comfortable when you ask them for help or for information than when you just go up and start talking about things in general.
Remember that you're out actually to make acquaintances! Friendships are fewer and farther between, and take a lot more time to develop. You don't need to be desperate, and you don't need to wonder what they think of you - just be interested in them.
When I get really terror-stricken, I pretend to be my friend Cindy, who is very confident and outgoing, and I do what I know she would do. Once I get started, I can be me again.